How to get from one format to another.

[Wynn]

Hi,

I am a University student studying Physics and need help understanding how you can go from an equation in format:

To:


For those curious I am looking at an explanation of time dilation where proof of the Lorentz gamma factor is provided. In the above example:
a represents the change in time
b represents the distance
c represents the speed of light (3.00x10^8 m/s)
and d represents velocity of the moving intertial frame as viewed from outside.

(Yes, I know I am not using standard symbols to notate the correct things like using v for velocity, delta t for change in time and so on. I am merely asking the question in a general form of how to get from the first shown equation to the second.)

 

[Deleted member 4993]

Hi,

I am a University student studying Physics and need help understanding how you can go from an equation in format:
View attachment 4505
To:
View attachment 4504

For those curious I am looking at an explanation of time dilation where proof of the Lorentz gamma factor is provided. In the above example:
a represents the change in time
b represents the distance
c represents the speed of light (3.00x10^8 m/s)
and d represents velocity of the moving intertial frame as viewed from outside.

(Yes, I know I am not using standard symbols to notate the correct things like using v for velocity, delta t for change in time and so on. I am merely asking the question in a general form of how to get from the first shown equation to the second.)

c² - d² = c² * (1 - d²/c²)

√[c² - d²] = √[c² * (1 - d²/c²)] = √c² * √(1 - d²/c²) = c * √(1 - d²/c²)

 

[Dale10101]

Using words

Hi,

I am a University student studying Physics and need help understanding how you can go from an equation in format:
View attachment 4505
To:
View attachment 4504

For those curious I am looking at an explanation of time dilation where proof of the Lorentz gamma factor is provided. In the above example:
a represents the change in time
b represents the distance
c represents the speed of light (3.00x10^8 m/s)
and d represents velocity of the moving intertial frame as viewed from outside.

(Yes, I know I am not using standard symbols to notate the correct things like using v for velocity, delta t for change in time and so on. I am merely asking the question in a general form of how to get from the first shown equation to the second.)

To add no new information except the overview; you can extract a factor from beneath the radical sign if its exponent matches the index of the radical.

As was demonstrated, the first step was factoring c². You could also factor out c⁴ and extract it using c⁴ = ((c²)²) ... you could also factor and extract, d² or (cd)². The point is, that you can float things out from beneath the radical by a two step process that begins with factoring with respect to the radicals index. The other point to be aware of is that if the index is even then the extracted factor must be put within the absolute value bars unless, as in this case, it is defined as a positive value.